L(s) = 1 | − 4-s + 6·11-s + 16-s − 4·19-s + 12·29-s + 10·31-s + 12·41-s − 6·44-s + 13·49-s + 24·59-s + 16·61-s − 64-s + 4·76-s − 16·79-s − 36·89-s + 6·101-s − 4·109-s − 12·116-s + 5·121-s − 10·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 1.80·11-s + 1/4·16-s − 0.917·19-s + 2.22·29-s + 1.79·31-s + 1.87·41-s − 0.904·44-s + 13/7·49-s + 3.12·59-s + 2.04·61-s − 1/8·64-s + 0.458·76-s − 1.80·79-s − 3.81·89-s + 0.597·101-s − 0.383·109-s − 1.11·116-s + 5/11·121-s − 0.898·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.657919071\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.657919071\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 193 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.656858818069400644858330271013, −9.579611049041271462256862232465, −8.741675086877256691838553666420, −8.734699258940243333176450954102, −8.270579265306059676889654749916, −8.087070634214029498384351773009, −7.03552846908471693451425525831, −7.00025529809323576744175469994, −6.67703276718226439866241869333, −6.00369298263222888751651844666, −5.78803637084768654825584545153, −5.22191986050896946834542710229, −4.43363282966278761210274107648, −4.17304952892473417713261866067, −4.13875420892722832122800955967, −3.26428706635492863075742539651, −2.62886704389196821115811099912, −2.21592648694694688253795068334, −1.07450360904072939435424155074, −0.885440186468675407454991193680,
0.885440186468675407454991193680, 1.07450360904072939435424155074, 2.21592648694694688253795068334, 2.62886704389196821115811099912, 3.26428706635492863075742539651, 4.13875420892722832122800955967, 4.17304952892473417713261866067, 4.43363282966278761210274107648, 5.22191986050896946834542710229, 5.78803637084768654825584545153, 6.00369298263222888751651844666, 6.67703276718226439866241869333, 7.00025529809323576744175469994, 7.03552846908471693451425525831, 8.087070634214029498384351773009, 8.270579265306059676889654749916, 8.734699258940243333176450954102, 8.741675086877256691838553666420, 9.579611049041271462256862232465, 9.656858818069400644858330271013